Properties

Label 600.1.q.b.107.1
Level 600
Weight 1
Character 600.107
Analytic conductor 0.299
Analytic rank 0
Dimension 8
Projective image \(D_{6}\)
CM discriminant -8
Inner twists 16

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Newspace parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 600.q (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.299439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.5400000.2

Embedding invariants

Embedding label 107.1
Root \(0.258819 - 0.965926i\) of \(x^{8} - x^{4} + 1\)
Character \(\chi\) \(=\) 600.107
Dual form 600.1.q.b.443.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{2} +(-0.258819 + 0.965926i) q^{3} -1.00000i q^{4} +(-0.500000 - 0.866025i) q^{6} +(0.707107 + 0.707107i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{2} +(-0.258819 + 0.965926i) q^{3} -1.00000i q^{4} +(-0.500000 - 0.866025i) q^{6} +(0.707107 + 0.707107i) q^{8} +(-0.866025 - 0.500000i) q^{9} +1.73205i q^{11} +(0.965926 + 0.258819i) q^{12} -1.00000 q^{16} +(-0.707107 + 0.707107i) q^{17} +(0.965926 - 0.258819i) q^{18} +1.00000i q^{19} +(-1.22474 - 1.22474i) q^{22} +(-0.866025 + 0.500000i) q^{24} +(0.707107 - 0.707107i) q^{27} +(0.707107 - 0.707107i) q^{32} +(-1.67303 - 0.448288i) q^{33} -1.00000i q^{34} +(-0.500000 + 0.866025i) q^{36} +(-0.707107 - 0.707107i) q^{38} -1.73205i q^{41} +1.73205 q^{44} +(0.258819 - 0.965926i) q^{48} +1.00000i q^{49} +(-0.500000 - 0.866025i) q^{51} +1.00000i q^{54} +(-0.965926 - 0.258819i) q^{57} +1.00000i q^{64} +(1.50000 - 0.866025i) q^{66} +(1.22474 + 1.22474i) q^{67} +(0.707107 + 0.707107i) q^{68} +(-0.258819 - 0.965926i) q^{72} +(1.22474 - 1.22474i) q^{73} +1.00000 q^{76} +(0.500000 + 0.866025i) q^{81} +(1.22474 + 1.22474i) q^{82} +(-0.707107 - 0.707107i) q^{83} +(-1.22474 + 1.22474i) q^{88} +1.73205 q^{89} +(0.500000 + 0.866025i) q^{96} +(-0.707107 - 0.707107i) q^{98} +(0.866025 - 1.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{6} + O(q^{10}) \) \( 8q - 4q^{6} - 8q^{16} - 4q^{36} - 4q^{51} + 12q^{66} + 8q^{76} + 4q^{81} + 4q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(3\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(4\) 1.00000i 1.00000i
\(5\) 0 0
\(6\) −0.500000 0.866025i −0.500000 0.866025i
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(9\) −0.866025 0.500000i −0.866025 0.500000i
\(10\) 0 0
\(11\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(18\) 0.965926 0.258819i 0.965926 0.258819i
\(19\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.22474 1.22474i −1.22474 1.22474i
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(25\) 0 0
\(26\) 0 0
\(27\) 0.707107 0.707107i 0.707107 0.707107i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.707107 0.707107i 0.707107 0.707107i
\(33\) −1.67303 0.448288i −1.67303 0.448288i
\(34\) 1.00000i 1.00000i
\(35\) 0 0
\(36\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) −0.707107 0.707107i −0.707107 0.707107i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 1.73205 1.73205
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0.258819 0.965926i 0.258819 0.965926i
\(49\) 1.00000i 1.00000i
\(50\) 0 0
\(51\) −0.500000 0.866025i −0.500000 0.866025i
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 1.00000i 1.00000i
\(55\) 0 0
\(56\) 0 0
\(57\) −0.965926 0.258819i −0.965926 0.258819i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 1.50000 0.866025i 1.50000 0.866025i
\(67\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(68\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −0.258819 0.965926i −0.258819 0.965926i
\(73\) 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.00000 1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(82\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(83\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −1.22474 + 1.22474i −1.22474 + 1.22474i
\(89\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) −0.707107 0.707107i −0.707107 0.707107i
\(99\) 0.866025 1.50000i 0.866025 1.50000i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(108\) −0.707107 0.707107i −0.707107 0.707107i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(114\) 0.866025 0.500000i 0.866025 0.500000i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −2.00000
\(122\) 0 0
\(123\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) −0.707107 0.707107i −0.707107 0.707107i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) −0.448288 + 1.67303i −0.448288 + 1.67303i
\(133\) 0 0
\(134\) −1.73205 −1.73205
\(135\) 0 0
\(136\) −1.00000 −1.00000
\(137\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(138\) 0 0
\(139\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(145\) 0 0
\(146\) 1.73205i 1.73205i
\(147\) −0.965926 0.258819i −0.965926 0.258819i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(153\) 0.965926 0.258819i 0.965926 0.258819i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.965926 0.258819i −0.965926 0.258819i
\(163\) 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(164\) −1.73205 −1.73205
\(165\) 0 0
\(166\) 1.00000 1.00000
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) 0 0
\(171\) 0.500000 0.866025i 0.500000 0.866025i
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.73205i 1.73205i
\(177\) 0 0
\(178\) −1.22474 + 1.22474i −1.22474 + 1.22474i
\(179\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.22474 1.22474i −1.22474 1.22474i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −0.965926 0.258819i −0.965926 0.258819i
\(193\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0.448288 + 1.67303i 0.448288 + 1.67303i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(202\) 0 0
\(203\) 0 0
\(204\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.73205 −1.73205
\(210\) 0 0
\(211\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.00000i 1.00000i
\(215\) 0 0
\(216\) 1.00000 1.00000
\(217\) 0 0
\(218\) 0 0
\(219\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1.00000 −1.00000
\(227\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 1.41421 1.41421i 1.41421 1.41421i
\(243\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(244\) 0 0
\(245\) 0 0
\(246\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.866025 0.500000i 0.866025 0.500000i
\(250\) 0 0
\(251\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) −0.866025 1.50000i −0.866025 1.50000i
\(265\) 0 0
\(266\) 0 0
\(267\) −0.448288 + 1.67303i −0.448288 + 1.67303i
\(268\) 1.22474 1.22474i 1.22474 1.22474i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0.707107 0.707107i 0.707107 0.707107i
\(273\) 0 0
\(274\) 1.00000i 1.00000i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) −1.22474 1.22474i −1.22474 1.22474i
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0.866025 0.500000i 0.866025 0.500000i
\(295\) 0 0
\(296\) 0 0
\(297\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.00000i 1.00000i
\(305\) 0 0
\(306\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(307\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(322\) 0 0
\(323\) −0.707107 0.707107i −0.707107 0.707107i
\(324\) 0.866025 0.500000i 0.866025 0.500000i
\(325\) 0 0
\(326\) 1.73205i 1.73205i
\(327\) 0 0
\(328\) 1.22474 1.22474i 1.22474 1.22474i
\(329\) 0 0
\(330\) 0 0
\(331\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(338\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(339\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(340\) 0 0
\(341\) 0 0
\(342\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(353\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.73205i 1.73205i
\(357\) 0 0
\(358\) 1.22474 1.22474i 1.22474 1.22474i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 0.517638 1.93185i 0.517638 1.93185i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(368\) 0 0
\(369\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 1.73205 1.73205
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0.866025 0.500000i 0.866025 0.500000i
\(385\) 0 0
\(386\) 1.73205i 1.73205i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.50000 0.866025i −1.50000 0.866025i
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0.448288 1.67303i 0.448288 1.67303i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0.258819 0.965926i 0.258819 0.965926i
\(409\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(410\) 0 0
\(411\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(418\) 1.22474 1.22474i 1.22474 1.22474i
\(419\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0.707107 0.707107i 0.707107 0.707107i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.707107 0.707107i −0.707107 0.707107i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(433\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −1.67303 0.448288i −1.67303 0.448288i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0.500000 0.866025i 0.500000 0.866025i
\(442\) 0 0
\(443\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(450\) 0 0
\(451\) 3.00000 3.00000
\(452\) 0.707107 0.707107i 0.707107 0.707107i
\(453\) 0 0
\(454\) 2.00000i 2.00000i
\(455\) 0 0
\(456\) −0.500000 0.866025i −0.500000 0.866025i
\(457\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(458\) 0 0
\(459\) 1.00000i 1.00000i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −2.00000 −2.00000
\(467\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(483\) 0 0
\(484\) 2.00000i 2.00000i
\(485\) 0 0
\(486\) 0.500000 0.866025i 0.500000 0.866025i
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) 0 0
\(489\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0.448288 1.67303i 0.448288 1.67303i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(499\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(513\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(514\) 2.00000i 2.00000i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −0.866025 1.50000i −0.866025 1.50000i
\(535\) 0 0
\(536\) 1.73205i 1.73205i
\(537\) 0.448288 1.67303i 0.448288 1.67303i
\(538\) 0 0
\(539\) −1.73205 −1.73205
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.00000i 1.00000i
\(545\) 0 0
\(546\) 0 0
\(547\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(548\) −0.707107 0.707107i −0.707107 0.707107i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.00000 −1.00000
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 1.50000 0.866025i 1.50000 0.866025i
\(562\) 0 0
\(563\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.73205i 1.73205i
\(567\) 0 0
\(568\) 0 0
\(569\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.500000 0.866025i 0.500000 0.866025i
\(577\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(578\) 0 0
\(579\) −0.866025 1.50000i −0.866025 1.50000i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.73205 1.73205
\(585\) 0 0
\(586\) 0 0
\(587\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(588\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(594\) −1.73205 −1.73205
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 0 0
\(603\) −0.448288 1.67303i −0.448288 1.67303i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(608\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −0.258819 0.965926i −0.258819 0.965926i
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 1.73205 1.73205
\(615\) 0 0
\(616\) 0 0
\(617\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.448288 1.67303i 0.448288 1.67303i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0.258819 0.965926i 0.258819 0.965926i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −0.965926 0.258819i −0.965926 0.258819i
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.00000 1.00000
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.22474 1.22474i −1.22474 1.22474i
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.73205i 1.73205i
\(657\) −1.67303 + 0.448288i −1.67303 + 0.448288i
\(658\) 0 0
\(659\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(663\) 0 0
\(664\) 1.00000i 1.00000i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) −1.73205 −1.73205
\(675\) 0 0
\(676\) −1.00000 −1.00000
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0.258819 0.965926i 0.258819 0.965926i
\(679\) 0 0
\(680\) 0 0
\(681\) −1.00000 1.73205i −1.00000 1.73205i
\(682\) 0 0
\(683\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(684\) −0.866025 0.500000i −0.866025 0.500000i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.00000i 1.00000i
\(695\) 0 0
\(696\) 0 0
\(697\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(698\) 0 0
\(699\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.73205 −1.73205
\(705\) 0 0
\(706\) 2.00000 2.00000
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.73205i 1.73205i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(724\) 0 0
\(725\) 0 0
\(726\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.12132 + 2.12132i −2.12132 + 2.12132i
\(738\) −0.448288 1.67303i −0.448288 1.67303i
\(739\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(748\) −1.22474 + 1.22474i −1.22474 + 1.22474i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(759\) 0 0
\(760\) 0 0
\(761\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(769\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(770\) 0 0
\(771\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(772\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.73205 1.73205
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000i 1.00000i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 1.67303 0.448288i 1.67303 0.448288i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.50000 0.866025i −1.50000 0.866025i
\(802\) −1.22474 1.22474i −1.22474 1.22474i
\(803\) 2.12132 + 2.12132i 2.12132 + 2.12132i
\(804\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0